N12 - Fractions and percentages of amounts | AQA GCSE Mathematics

A fraction or percentage can tell you what to do to an amount: take half, take three fifths, or take thirty per cent. You will use these instructions as multipliers and choose the right whole when a problem involves a group inside a group.

Fractions as operators

There are $240$ pupils in a year group. Three fifths of them cycle to school, and one quarter of those cyclists wear a reflective band. How many cyclists wear a band? The word “of” appears twice: each time, it tells you which amount to work on. We will return to the second step after making the first one secure.

You already know

A denominator tells you how many equal parts make one whole; a numerator tells you how many of those parts to take. Multiplying by a fraction means dividing by its denominator and multiplying by its numerator.

To find $\frac{3}{5}$ of $240$ , divide $240$ into five equal parts. Each part is $48$ . Taking three of those parts gives $144$ . Here the fraction is an instruction that acts on the amount.

Operator

A rule that acts on an input amount to produce an output amount. As a fraction operator, $\frac{3}{5}$ means “multiply by $\frac{3}{5}$ ”. For example, $\frac{3}{5}$ of $240$ is $144$ . It does not mean subtract $\frac{3}{5}$ from $240$ .

You can multiply first and divide afterwards, but dividing first often keeps the numbers smaller. Both orders give the same answer.

A fraction of an amount

For a whole-number denominator $b > 0$ ,

\frac{a}{b} \text{ of an amount} = \text{amount} \div b \times a

The amount being divided is the whole named after “of”.

Not on the formulae sheet: learn it.

The first example uses a proper fraction, so the answer should be smaller than the positive amount you start with.

Worked example 1: taking three equal parts

Question. Work out $\frac{3}{5}$ of $240$ . Do not use a calculator.

Plan. Find one fifth by dividing by $5$ , then take three fifths.

\begin{aligned} \frac{1}{5} \text{ of } 240 &= 240 \div 5 = 48 && \text{one equal part} \\ \frac{3}{5} \text{ of } 240 &= 48 \times 3 && \text{take three parts} \\ &= 144 && \text{multiply} \end{aligned}

Why this step? Divide by the denominator, $5$ , because five equal parts make the whole. Dividing by the numerator would make the wrong-sized parts.

Answer. $144$

Check. Three fifths is more than a half and less than one whole. The answer $144$ is between $120$ and $240$ .

The next example changes only the numerator, from $3$ to $7$ . There are now more than five fifths, so the output is larger than the input.

Worked example 2: an improper fraction operator

Question. Work out $\frac{7}{5}$ of $240$ . Do not use a calculator.

Plan. Keep the same fifth, $48$ , and take seven of them.

\begin{aligned} \frac{1}{5} \text{ of } 240 &= 240 \div 5 = 48 && \text{one fifth} \\ \frac{7}{5} \text{ of } 240 &= 48 \times 7 && \text{take seven fifths} \\ &= 336 && \text{multiply} \end{aligned}

Why this step? An operator greater than $1$ makes a positive amount larger. The word “fraction” does not mean the answer must be smaller.

Answer. $336$

Check. $\frac{7}{5} = 1 + \frac{2}{5}$ , so the answer is $240 + 96 = 336$ .

Compare the fraction with $1$ before doing any arithmetic. It gives you a useful size check.

Watch out

Wrong: $\frac{3}{5}$ of $240 = 240 \div 3 \times 5 = 400$ . This reverses the two jobs. Right: $240 \div 5 \times 3 = 144$ .

The next amount has the same denominator but a different number of parts to take.

Percentages without a calculator

A percentage is a fraction with denominator $100$ , so it can also act on an amount.

Percentage

“Per cent” means “out of one hundred”. As an operator, $35\%$ means multiply by $\frac{35}{100}$ , which is $0.35$ . For example, $50\%$ of an amount means half of it, not $50$ added to it.

Start with easy pieces. To find $10\%$ , divide by $10$ ; to find $1\%$ , divide by $100$ . Halving $10\%$ gives $5\%$ . These pieces combine to make less familiar percentages without a calculator.

Useful percentage pieces

For any amount,

\begin{aligned} 10\% \text{ of the amount} &= \text{amount} \div 10 \\ 1\% \text{ of the amount} &= \text{amount} \div 100 \\ 5\% \text{ of the amount} &= \text{amount} \div 20 \end{aligned}

The last line divides by $10$ , then by $2$ : it halves $10\%$ .

Not on the formulae sheet: learn it.

Choose pieces that add to the requested percentage. Work out their amounts before adding them.

Worked example 3: building thirty-five per cent

Question. Work out $35\%$ of $240$ . Do not use a calculator.

Plan. Use $35\% = 30\% + 5\%$ , building both pieces from $10\%$ .

\begin{aligned} 10\% \text{ of } 240 &= 240 \div 10 = 24 && \text{one tenth} \\ 30\% \text{ of } 240 &= 24 \times 3 = 72 && \text{three tenths} \\ 5\% \text{ of } 240 &= 24 \div 2 = 12 && \text{half of ten per cent} \\ 35\% \text{ of } 240 &= 72 + 12 = 84 && \text{combine the pieces} \end{aligned}

Why this step? $5\%$ is half of $10\%$ , so halve $24$ , not $240$ . Halving the original amount would give $50\%$ .

Answer. $84$

Check. $35\%$ is between $30\%$ and $40\%$ . The answer $84$ is between $72$ and $96$ .

The next example changes only the percentage to $65\%$ . Keep the same $10\%$ and $5\%$ pieces; choose a different multiple of $10\%$ .

Worked example 4: reusing the same pieces

Question. Work out $65\%$ of $240$ . Do not use a calculator.

Plan. Add $60\%$ and $5\%$ of the same amount.

\begin{aligned} 10\% \text{ of } 240 &= 24 && \text{divide by } 10 \\ 60\% \text{ of } 240 &= 24 \times 6 = 144 && \text{six lots of ten per cent} \\ 5\% \text{ of } 240 &= 24 \div 2 = 12 && \text{halve ten per cent} \\ 65\% \text{ of } 240 &= 144 + 12 = 156 && \text{add the required pieces} \end{aligned}

Why this step? Both percentages refer to the original $240$ . The $5\%$ is not taken from the $60\%$ amount.

Answer. $156$

Check. $35\% + 65\% = 100\%$ , and $84 + 156 = 240$ .

The arithmetic must follow the meaning of the percentage, rather than the digits in its name.

Watch out

Wrong: $5\%$ of $240 = 240 \div 5 = 48$ . That calculation finds one fifth, which is $20\%$ . Right: $5\%$ is half of $10\%$ , so it is $24 \div 2 = 12$ .

Show your easy pieces in a non-calculator solution. A correct method can still earn marks if the final addition slips.

Percentage multipliers

A decimal multiplier takes the place of all those separate percentage pieces in one multiplication.

For $37\%$ of $240$ , you need $\frac{37}{100}$ of $240$ . The fraction $\frac{37}{100}$ equals $0.37$ , so multiply by $0.37$ . The $0.37$ is the multiplier: the number by which you multiply the original amount.

A percentage of an amount

For a percentage $p\%$ ,

p\% \text{ of an amount} = \frac{p}{100} \times \text{amount}

The multiplier is $p \div 100$ . This also works when $p$ is a decimal, such as $37.5$ .

Not on the formulae sheet: learn it.

There is a neat reversal hidden in this rule: $35\%$ of $80$ and $80\%$ of $35$ are both $28$ . Each is $35 \times 80 \div 100$ . Swapping the two numbers leaves the product unchanged. It can turn an awkward calculation into an easy one.

Worked example 5: using a decimal multiplier

Question. A mixture has mass $240\,\mathrm{g}$ . $37\%$ of its mass is flour. Work out the mass of flour. A calculator may be used.

Plan. Divide the percentage by $100$ , then multiply the full mass by that decimal.

\begin{aligned} 37 \div 100 &= 0.37 && \text{the multiplier} \\ 0.37 \times 240 &= 88.8 && \text{calculator expression and display} \end{aligned}

Why this step? Multiplying by $0.37$ takes $37$ hundredths of the mass. Multiplying by $37$ would give $100$ times too much.

Answer. $88.8\,\mathrm{g}$

Check. $40\%$ of $240$ is $96$ , so $88.8$ is a sensible slightly smaller answer.

The next example changes only the percentage, to one with a decimal part. Dividing by $100$ still means moving the digits two places right.

Worked example 6: a decimal percentage

Question. A mixture has mass $240\,\mathrm{g}$ . $37.5\%$ of its mass is flour. Work out the mass of flour. A calculator may be used.

Plan. Use $37.5 \div 100$ as the multiplier, keeping all its decimal places.

\begin{aligned} 37.5 \div 100 &= 0.375 && \text{the multiplier} \\ 0.375 \times 240 &= 90 && \text{calculator expression and display} \end{aligned}

Why this step? The percentage has a decimal already, but it still needs dividing by $100$ . The multiplier is $0.375$ , not $3.75$ or $0.0375$ .

Answer. $90\,\mathrm{g}$

Check. $0.375 = \frac{3}{8}$ , and $240 \div 8 \times 3 = 90$ .

You can use fraction, percentage or decimal notation for the same operator. Choose the form that makes the calculation easiest.

Watch out

Wrong: $0.5\% = 0.5$ . Right: $0.5\% = 0.5 \div 100 = 0.005$ . Half a per cent is one two-hundredth of an amount; $0.5$ of it is a half.

Before pressing equals, compare the multiplier with $1$ . These percentages are less than $100\%$ , so the mass of flour must be less than the full mass.

Taking a fraction or percentage of a group

In a problem with several groups, the most important choice is the amount that each operator acts on.

Return to the opening year group. First, $\frac{3}{5}$ acts on all $240$ pupils and gives $144$ cyclists. Then $\frac{1}{4}$ , or $25\%$ , acts on those $144$ cyclists. It does not act on the whole year group.

[DIAGRAM: asset_name: Two operators and two different inputs; asset_slug: aqa8300_n21_subgroup_operators; caption: The second operator acts on 144 cyclists, not on all 240 pupils; description: Three boxes connected left to right. The first says all pupils, 240. An arrow labelled times three fifths leads to a blue box saying cyclists, 144. A second arrow labelled times one quarter, also 25 per cent, leads to a sage box saying cyclists with bands, 36. Beneath the first arrow the text reads first input 240, while beneath the second it reads second input 144.]
The second operator acts on 144 cyclists, not on all 240 pupils

The steps can be written as $240 \times \frac{3}{5} \times \frac{1}{4}$ . It is often clearer to calculate and label the intermediate amount first. For positive fractions smaller than $1$ , the final subgroup must be smaller than either group it came from.

Worked example 7: a quarter of the cyclists

Question. There are $240$ pupils in a year group. $\frac{3}{5}$ cycle to school. $25\%$ of the cyclists wear a reflective band. Work out how many cyclists wear a reflective band. Do not use a calculator.

Plan. Find the cyclists first, then take a quarter of that group.

\begin{aligned} 240 \div 5 \times 3 &= 144 && \text{pupils who cycle} \\ 25\% \text{ of } 144 &= 144 \div 4 && \text{a quarter of the cyclists} \\ &= 36 && \text{cyclists with bands} \end{aligned}

Why this step? “Of the cyclists” names the input for $25\%$ . Using $240$ again would count a quarter of the wrong group.

Answer. $36$ cyclists wear a reflective band.

Check. $36 \times 4 = 144$ , and $144 \div 3 \times 5 = 240$ , so both stages reverse correctly.

The next example changes only which group the percentage describes. You must now find the pupils who do not cycle before applying $25\%$ .

Worked example 8: a quarter of the other group

Question. There are $240$ pupils in a year group. $\frac{3}{5}$ cycle to school. $25\%$ of the pupils who do not cycle wear a reflective band. Work out how many pupils who do not cycle wear a reflective band. Do not use a calculator.

Plan. Subtract the cyclists from the whole year group, then take a quarter of the remaining pupils.

\begin{aligned} 240 \div 5 \times 3 &= 144 && \text{pupils who cycle} \\ 240 - 144 &= 96 && \text{pupils who do not cycle} \\ 96 \div 4 &= 24 && \text{a quarter of that group} \end{aligned}

Why this step? The percentage is unchanged, but its input is now $96$ , not $144$ . A percentage on its own does not tell you an amount.

Answer. $24$ pupils who do not cycle wear a reflective band.

Check. $144 + 96 = 240$ , and $24 \times 4 = 96$ .

Adding the two operator values does not apply one to the result of the other.

Watch out

Wrong: a quarter of three fifths of a group means taking $\frac{1}{4} + \frac{3}{5}$ of it. Right: take $\frac{3}{5}$ first, then $\frac{1}{4}$ of that result. The combined fraction is $\frac{1}{4} \times \frac{3}{5} = \frac{3}{20}$ .

Before calculating the final question, underline the words that name the input for $25\%$ .

Quick Recap

A fraction operator divides an amount into equal parts, then takes the required number of parts.
Build non-calculator percentages from useful pieces such as $10\%$ , $5\%$ and $1\%$ .
The multiplier for $p\%$ of an amount is $p \div 100$ .
Each “of” names an input. A percentage of a subgroup acts on that subgroup, not automatically on the original whole.

In the opening year group, $\frac{3}{5}$ of $240$ is $144$ cyclists, and $\frac{1}{4}$ of $144$ is $36$ cyclists with bands.

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N12 - Fractions and percentages of amounts

Fractions as operators

Percentages without a calculator

Percentage multipliers

Taking a fraction or percentage of a group

Quick Recap

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